By the assumption on \(L\) , \(\widetilde{S}_n\)  is aperiodic, therefore from [1]},
it follows that
\(\varrho := \sup \lbrace  |\psi (\varphi )|: \varphi \in [-\pi ,\pi ]^d, {\left\Vert \varphi  \right\Vert }> r \rbrace <1.\) 
